Although sensitivity analysis provides valuable information for structural optimization, it is often difficult to use the Hessian in large models since many methods still suffer from inaccuracy, inefficiency, or limitation issues. In this context, we report the theoretical description of a general sensitivity procedure that calculates the diagonal terms of the Hessian matrix by using a new variant of hyper-dual numbers as derivative tool. We develop a diagonal variant of hyper-dual numbers and their arithmetic to obtain the exact derivatives of tensor-valued functions of a vector argument, which comprise the main contributions of this work. As this differentiation scheme represents a general black-box tool, we supply the computer implementation of the hyper-dual formulation in Fortran. By focusing on the diagonal terms, the proposed sensitivity scheme is significantly lighter in terms of computational costs, facilitating the application in engineering problems. As an additional strategy to improve efficiency, we highlight that we perform the derivative calculation at the element-level. This work can contribute to many studies since the sensitivity scheme can adapt itself to numerous finite element formulations or problem settings. The proposed method promotes the usage of second-order optimization algorithms, which may allow better convergence rates to solve intricate problems in engineering applications.

中文翻译：

---

使用对角超对偶数进行二阶设计灵敏度分析

尽管敏感性分析为结构优化提供了有价值的信息，但通常很难在大型模型中使用 Hessian，因为许多方法仍然存在不准确、低效或限制问题。在这种情况下，我们报告了一般敏感性程序的理论描述，该程序通过使用超对偶数的新变体作为导数工具来计算 Hessian 矩阵的对角项。我们开发了超对偶数及其算术的对角线变体，以获得向量参数的张量值函数的精确导数，这构成了这项工作的主要贡献。由于这个微分方案代表了一个通用的黑盒工具，我们提供了 Fortran 中超对偶公式的计算机实现。通过关注对角线项，所提出的灵敏度方案在计算成本方面明显更轻，有利于在工程问题中的应用。作为提高效率的附加策略，我们强调我们在元素级别执行导数计算。这项工作可以为许多研究做出贡献，因为灵敏度方案可以适应许多有限元公式或问题设置。所提出的方法促进了二阶优化算法的使用，这可能允许更好的收敛速度来解决工程应用中的复杂问题。这项工作可以为许多研究做出贡献，因为灵敏度方案可以适应许多有限元公式或问题设置。所提出的方法促进了二阶优化算法的使用，这可能允许更好的收敛速度来解决工程应用中的复杂问题。这项工作可以为许多研究做出贡献，因为灵敏度方案可以适应许多有限元公式或问题设置。所提出的方法促进了二阶优化算法的使用，这可能允许更好的收敛速度来解决工程应用中的复杂问题。